Basic Quantities of Interest - INTRODUCTION TO STRUCTURAL ANALYSIS

Chapter 1 INTRODUCTION TO STRUCTURAL ANALYSIS

1.5 Basic Quantities of Interest

This section devotes to describe two different classes of basic quantities that are involved in structural analysis, one is termed kinematical quantities and the other is termed static quantities.

1.5.1 Kinematical quantities

Kinematical quantities describe geometry of both the undeformed and deformed configurations of the structure. Within the context of static structural analysis, kinematical quantities can be categorized into two different sets: one associated with quantities used to measure the movement or change in position of the structure and the other is associated with quantities used to measure the change in shape or distortion of the structure.

Displacement at any point within the structure is a quantity representing the change in position of that point in the deformed configuration measured relative to the undeformed configuration. Rotation at any point within the structure is a quantity representing the change in orientation of that point in the deformed configuration measured relative to the undeformed configuration. For a plane structure shown in Figure 1.18, the displacement at any point is fully described by a two-component vector (u, v) where u is a component of the displacement in X- direction and v is a component of the displacement in Y-direction while the rotation at any point is fully described by an angle  measured from a local tangent line in the undeformed configuration to a local tangent line at the same point in the deformed configuration. It is important to emphasize that the rotation is not an independent quantity but its value at any point can be computed when the displacement at that point and all its neighboring points is known.

A degree of freedom, denoted by DOF, is defined as a component of the displacement or the rotation at any node (of the discrete structure) essential for describing the displacement of the entire structure. There are two types of the degree of freedom, one termed as a prescribed degree of freedom and the other termed as a free or unknown degree of freedom. The former is the degree of freedom that is known a priori, for instance, the degree of freedom at nodes located at supports where components of the displacement or rotation are known while the latter is the degree of freedom that is unknown a priori. The number of degrees of freedom at each node depends primarily on the type of nodes and structures and also the internal releases and constraints present within the structure. In general, it is equal to the number of independent degrees of freedom at that node essential for describing the displacement of the entire structure. For beams, plane trusses,

space truss, plane frames, and space frames containing no internal release and constraint, the number of degrees of freedom per node are 2 (a vertical displacement and a rotation), 2 (two components of the displacement), 3 (three components of the displacement), 3 (two components of the displacement and a rotation) and 6 (three components of the displacement and three components of the rotation), respectively. Figure 1.21 shows examples of both prescribed degrees of freedom and free degrees of freedom of beam, plane truss and plane frames. The number of degrees of freedom of a structure is defined as the number of all independent degrees of freedom sufficient for describing the displacement of the entire structure or, equivalently, it is equal to the sum of numbers of degrees of freedom at all nodes. For instance, a beam shown in Figure 1.21(a) has 6 DOFs {v1, 1, v2, 2, v3, 3} consisting of 3 prescribed DOFs {v1, 1, v3} and 3 free DOFs {v2, 2, 3}; a plane truss shown in Figure 1.21(b) has 6 DOFs {u1, v1, u2, v2, u3, v3} consisting of 3 prescribed DOFs {u1, v1, v2} and 3 free DOFs {u2, u3, v3}; and a plane frame shown in Figure 1.21(c) has 9 DOFs {u1, v1, 1, u2, v2, 2, u3, v3, 3} consisting of 3 prescribed DOFs { u1, v1, v3} and 6 free DOFs {1, u2, v2, 2, u3, 3}. It is evident that the number of degrees of freedom of a given structure is not unique but depending primarily on how the structure is discretized. As the number of nodes in the discrete structure increases, the number of the degrees of freedom of the structure increases.

Figure 1.21: (a) Degrees of freedom of a beam, (b) degrees of freedom of a plane truss, and (c) degrees of freedom of a plane frame

X v1=0 u1=0 u3 v3 u2 v2=0 Y Node 1 Node 2 Node 3 (b) (c) (a) v1 = 0 1 = 0 v2 2 v3 = 0 3 Y X

Node 1 Node 2 Node 3

v1=0 u1=0 1 3 u3 v3=0 u2 v2 2 Y X Node 1 Node 2 Node 3

Deformation is a quantity used to measure the change in shape or the distortion of a structure (i.e. elongation, rate of twist, curvature, strain, etc.) due to disturbances and excitations. The deformation is a relative quantity and a primary source that produces the internal forces or stresses within the structure. For continuous structures, the deformation is said to be completely described if and only if the deformation is known at all points or is given as a function of position while, for discrete structures, the deformation of the entire structure is said to be completely described if and only if the deformation of all members constituting the structure are known. The deformation for each member of a discrete structure can be described by a finite number of quantities called the member deformation (this, however, must be furnished by certain assumptions on kinematics of the member to ensure that the deformation at every point within the member can be determined in terms of the member deformation). The quantities selected to be the member deformation depend primarily on the type and behavior of such member. For instance, the

elongation, e, or a measure of the change in length of a member is commonly chosen as the member deformation of a truss member as shown in Figure 1.22(a); the relative end rotations {s, e} where s and e denotes the rotations at both ends of the member measured relative to a chord connecting both end points as shown in Figure 1.22(b) are commonly chosen as the member deformation of a beam member; and the elongation and two relative end rotations {e, s, e} as shown in Figure 1.22(c) are commonly chosen as the member deformation of a frame member. It is remarked that the deformation of the entire discrete structure can fully be described by a finite set containing all member deformation.

Figure 1.22: Member deformation for different types of members: (a) truss member, (b) beam member, and (c) frame member

A Rigid body motion is a particular type of displacement that produces no deformation at any point within the structure. The rigid body motion can be decomposed into two parts: a rigid translation and a rigid rotation. The rigid translation produces the same displacement at all points while the rigid rotation produces the displacement that is a linear function of position. Figure 1.23 shows a plane structure undergoing a series of rigid body motions starting from a rigid translation in the X-direction, then a rigid translation in the Y-direction, and finally a rigid rotation about a point A´. L L´= L + e y x L L´= L y x e L L´= L+ e y x e (a) (b) (c) s s

Within the context of static structural analysis, the structure under consideration must sufficiently be constrained to prevent both the rigid body motion of the entire structure and the rigid body motion of any part of the structure. The former is prevented by providing a sufficient number of supports and proper directions against movement and the latter is prevented by the proper arrangement of members and their connections. A structure shown in Figure 1.24(a) is a structure in Figure 1.23 after prevented all possible rigid body motions by introducing a pinned support at a point A and a roller support at a point B. A structure shown in Figure 1.24(b) indicates that although many supports are provided but in improper manner, the structure can still experience the rigid body motion; for this particular structure, the rigid translation can still occur in the X- direction.

Figure 1.23: An unconstrained plane structure undergoing a series of rigid body motions

Figure 1.24: (a) A structure with sufficient constraints preventing all possible rigid body motions and (b) a structure with improper constraints

1.5.2 Static quantities

Quantities such as external actions and reactions in terms of forces and moments exerted to the structure by surrounding environments and the intensity of forces (e.g. stresses and pressure) and theirs resultants (e.g. axial force, bending moment, shear force, and torque, etc.) induced internally at any point within the structure are termed as static quantities. Applied load is one of static quantities referring to the prescribed force or moment acting to the structure. Support reaction is a term referring to an unknown force or moment exerted to the structure by idealized supports (representatives of surrounding environments) in order to prevent its movement or to maintain its

Y X (a) (b) Y X Y X A A´ B

stability. Support reactions are generally unknown a priori. There are two types of applied loads; one called a nodal load is an applied load acting to the node of the structure and the other called a

member loads is an applied load acting to the member. An example of applied loads (both nodal loads and member loads) and support reactions of a plane frame is depicted in Figure 1.25.

Stress is a static quantity used to describe the intensity of force (force per unit area) at any plane passing through a point. Internal force is a term used to represent the force or moment

resultant of stress components on a particular surface such as a cross section of a member. Note again that a major source that produces the stress and the internal force within the structure is the deformation. The distribution of both stress and internal force within the member depends primarily on characteristics or types of that member. For standard one-dimensional members in a plane structure such as an axial member, a flexural member, and a frame member, the internal force is typically defined in terms of the force and moment resultants of all stress components over the cross section of the member – a plane normal to the axis of the member.

Figure 1.25: Schematic of a plane frame subjected to external applied loads

An axial member is a member in which only one component of the internal force, termed as an axial force and denoted by f – a force resultant normal to the cross section, is present. The axial force f is considered positive if it results from a tensile stress present at the cross section; otherwise, it is considered negative. Figure 1.26 shows an axial member subjected to two forces {fx1, fx2} at its ends where fx1 and fx2 are considered positive if their directions are along the positive local x-axis. The axial force f at any cross section of the member can readily be related to the two end forces {fx1,

fx2} by enforcing static equilibrium of both parts of the member resulting from an imaginary cut; this gives rise to f = – fx1 = fx2. Such obtained relation implies that {f, fx1, fx2} are not all independent but only one of these three quantities can equivalently be chosen to fully represent the internal force of the axial member.

Figure 1.26: An axial member subjected to two end forces

A flexural member is a member in which only two components of the internal force, termed as a shear force denoted by V – a resultant force of the shear stress component and a bending

moment denoted by M – a resultant moment of the normal stress component, are present. The shear y x fx1 fx2 y x fx1 f f fx2 Node 1 Node 2 Node 3 Node 4

force V and the bending moment M are considered positive if their directions are as shown in Figure 1.27; otherwise, they are considered negative. Figure 1.27 illustrates a flexural member subjected to forces and moments {fy1, m1, fy2, m2} at its ends where fy1 and fy2 are considered positive if their directions are along the positive local y-axis and m1 and m2 are considered positive if their directions are along the positive local z-axis. The shear force V and the bending moment M at any cross section of the member can readily be related to the end forces and moments {fy1, m1, fy2, m2} by enforcing static equilibrium of both parts of the member resulting from an imaginary cut. It can be verified that only two quantities from a set {fy1, m1, fy2, m2} are independent and the rest can be obtained from equilibrium of the entire member. This implies in addition that two independent quantities from {fy1, m1, fy2, m2} can be chosen to fully represent the internal force of the flexural member; for instance, {m1, m2} is a common choice for the internal force of the flexural member.

Figure 1.27: A flexural member subjected to end forces and end moments.

A frame member is a member in which three components of the internal force (i.e. an axial force f, a shear force V, and a bending moment M) are present. The axial force f, the shear force V and the bending moment M are considered positive if their directions are as indicated in Figure 1.28; otherwise, they are considered negative. Figure 1.28 shows a frame member subjected to a set of forces and moments {fx1, fy1, m1, fx2, fy2, m2} at its ends where fx1 and fx2 are considered positive if their directions are along the positive local x-axis, fy1 and fy2 are considered positive if their directions are along the positive local y-axis and m1 and m2 are considered positive if their directions are along the positive local z-axis. The axial force can readily be related to the end forces {fx1, fx2} by a relation f = – fx1 = fx2 and the internal forces {V, M} at any cross section of the member can be related to the end forces and end moments {fy1, m1, fy2, m2} by enforcing static equilibrium to both parts of the member resulting from a cut. It can also be verified that only three quantities from a set {fx1, fy1, m1, fx2, fy2, m2} are independent and the rest can be obtained from equilibrium of the entire member. This implies that two independent quantities from {fy1, m1, fy2,

m2} along with one quantity from {f, fx1, fx2} can be chosen to fully represent the internal forces of the frame member; for instance, {f, m1, m2} is a common choice for the internal force of the frame member.

Figure 1.28: A frame member subjected to a set of end forces and end moments.

In document Fundamental Structural Analysis (2) (Page 30-35)